ts: Migrate pm_conversations module to typescript.

[Lalit3716]

• Migrates pm_conversations.js module to typescript.

Self-review checklist

• Self-reviewed the changes for clarity and maintainability
  (variable names, code reuse, readability, etc.).

Communicate decisions, questions, and potential concerns.

• Explains differences from previous plans (e.g., issue description).
• Highlights technical choices and bugs encountered.
• Calls out remaining decisions and concerns.
• Automated tests verify logic where appropriate.

Individual commits are ready for review (see commit discipline).

• Each commit is a coherent idea.
• Commit message(s) explain reasoning and motivation for changes.

Completed manual review and testing of the following:

• Visual appearance of the changes.
• Responsiveness and internationalization.
• Strings and tooltips.
• End-to-end functionality of buttons, interactions and flows.
• Corner cases, error conditions, and easily imagined bugs.

[Lalit3716]

This is ready for another look. @PIG208

[PIG208]

Thanks! This looks good to me. I have nothing else to add.

[timabbott]

Merged, thanks @Lalit3716!